Quadratic Realizability of Palindromic Matrix Polynomials: the Real Case

Abstract

Let \cL = (\cL_1,\cL_2) be a list consisting of structural data for a matrix polynomial; here \cL_1 is a sublist consisting of powers of irreducible (monic) scalar polynomials over the field \RR, and \cL_2 is a sublist of nonnegative integers. For an arbitrary such \cL, we give easy-to-check necessary and sufficient conditions for \cL to be the list of elementary divisors and minimal indices of some real $T$-palindromic quadratic matrix polynomial. For a list \cL satisfying these conditions, we show how to explicitly build a real $T$-palindromic quadratic matrix polynomial having \cL as its structural data; that is, we provide a $T$-palindromic quadratic realization of \cL over \RR. A significant feature of our construction differentiates it from related work in the literature; the realizations constructed here are direct sums of blocks with low bandwidth, that transparently display the spectral and singular structural data in the original list \cL